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Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

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A problem on intermediate value property and continuity

Let $f:\mathbb{R} \to \mathbb{R}$ be a function with the intermediate value property: that is, $f$ maps intervals to intervals. Let $x \in \mathbb{R}$. Suppose to each sequence $ (x_n) $ converging to $x$ there exists a constant $M$ such…
damini
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Show that $f$ cannot have infinitely many zeroes in $[0, 1]$.

Let $f : \mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f$ and its derivative have no common zero in the closed interval $[0, 1]$. Show that f cannot have infinitely many zeroes in $[0, 1]$.
Sriti Mallick
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Proof that all Cauchy sequences converge

I am supposed to prove that all Cauchy sequences converge, using the fact that a sequence converges iff its lim sup equals its lim inf. I think the proof I have works, but the one given in the solutions is different, so I'd like to make sure. Also,…
Liam
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What is the point of nuking this mosquito in Real Analysis by Shakarchi and Stein?

I have tried to read volume 3 of Shakarchi and Stein more than a few times now and I keep getting stuck in chapter one in the same place. After going through a bunch of basic concepts from analysis we finally come to a lemma, which says that if a…
Matt Calhoun
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No function continuous at rational points and discontinuous at irrational points in $[0,1]$

Let $C_f$ and $D_f$ mean sets where a function is continuous and discontinuous. I’m trying to prove there is no function $f:[0,1] \to \mathbb{R}$ such that $C_f = [0,1] \cap \mathbb{Q}$ and $D_f = [0,1] \setminus \mathbb{Q}$. I have seen a proof…
scobaco
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Using different characterizations of compactness, continuity, etc.

I've been looking for simple problems online to improve my grasp on the basics of elementary analysis. I'm not sure how much context I should include to make this question understood, so I'll just include it all; my question will be at the very…
Ryan
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Prove that if $S$ is a finite set then $S$ has no limit points.

Prove that if $S$ is a finite set then $S$ has no limit points. Can someone tell me if my approach is correct: Proof: Suppose $S$ is a finite set, then we can write $S = \{a_1, a_2, \ldots, a_n\}$ with $a_i \neq a_j$ if $i \neq j$. Suppose to the…
Robert Cardona
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Formula for the floor function

I found the following formula for the floor function: $$\lfloor x \rfloor = -\frac12+x+\frac{\arctan(\cot\pi x)}{\pi}$$ for all $x$ not an integer. My question is where I can find the proof of this formula.
DER
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Let $f\in L^1(\mathbb R)$ and let $F,G:\mathbb R\to\mathbb R$ be the functions defined by: ...

I have applied for a Ph.D. in Trieste and am preparing for the exams. I am having a problem with Problem 8 here. Here is the text. Let $f\in L^1(\mathbb R)$ and let $F,G:\mathbb R\to\mathbb R$ be the functions defined…
MickG
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Find all the continuous functions such that $\int_{-1}^{1}f(x)x^ndx=0$.

Find all the continuous functions on $[-1,1]$ such that $\int_{-1}^{1}f(x)x^ndx=0$ fof all the even integers $n$. Clearly, if $f$ is an odd function, then it satisfies this condition. What else?
user45955
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uniform convergence of few sequence of functions

Pick out the sequences $\{f_n\}$ which are uniformly convergent. (a) $f_n(x) = nxe^{−nx}$ on $(0,∞)$. (b)$f_n(x) = x^n$ on $[0, 1]$. (c)$f_n(x) = \frac{\sin(nx)}{\sqrt{n}}$ on $\mathbb{R}$. (d) $f_n(x)=\frac{nx}{1 + nx}$ on $(0,1)$ (e) $f_n(x) =…
poton
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proving derivative in real analysis

I have proved the following problem, can you help me check if there is any loopholes in my proof? Let I be an open interval in R, let $c \in I$, and let $f, g\colon I\to \mathbb{R}$ be functions. Suppose that $f(c) = g(c)$, and that $f(x) \leq g(x)$…
Lindsay Duran
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Prove that $f(x) = 0$ for all $x \in \mathbb{R}$ (Analysis)

Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) = f^{(4)}(x)$ with $f(0) = f’(0) = f’’(0) = f’’’(0) = 0.$ Prove $f(x) = 0$ for all $x \in \mathbb{R}$ My Attempts: Suppose $x \in \mathbb{R}$. Note that $\displaystyle f'(0) =…
Dominated Convergence Theorem
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A problem about convergence in finite measure space

Suppose we have a measure space $(X,M,\mu)$, where $\mu(X)<+\infty$. Let ${f_n}$ be a sequence of non-negative functions, $f_n\in L_1,\ \forall n$ and $f_n$ converges to some $f$ pointwise. $f$ is not necessarily in $L_1$. Suppose now…
qitao
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Real analysis question $e^{-1/x^2}$

Let $f$ be defined on $\mathbb{R}$ by $f(x) = e^{-1/x^2}$ for $x$ not equal to $0$. and $f(0)= 0$. Prove that $f^{(n)}(0)=0$ for all $n = 1, 2,3$ ... Do I need to use Taylor expansion from calculus class? Any hint would be appreciated.
Markus Xero
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